Find the following indefinite integral:∫((1−8𝑥)𝑥3)d𝑥=

Sure, let's solve the indefinite integral step by step:

$\int (1 - 8x)x^3 \, dx$

First, distribute $x^3$ inside the parentheses:

$\int (x^3 - 8x^4) \, dx$

Now, we can split the integral into two separate integrals:

$\int x^3 \, dx - \int 8x^4 \, dx$

Next, integrate each term separately. Recall that the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ plus a constant of integration $C$:

1. For $\int x^3 \, dx$:

$\int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4}$

2. For $\int 8x^4 \, dx$:

$\int 8x^4 \, dx = 8 \int x^4 \, dx = 8 \cdot \frac{x^{4+1}}{4+1} = 8 \cdot \frac{x^5}{5} = \frac{8x^5}{5}$

Now, combine the results:

$\int (x^3 - 8x^4) \, dx = \frac{x^4}{4} - \frac{8x^5}{5} + C$

So, the indefinite integral is:

$\int (1 - 8x)x^3 \, dx = \frac{x^4}{4} - \frac{8x^5}{5} + C$

where $C$ is the constant of integration.